Unified RKHS Methodology and Analysis for Functional Linear and Single-Index Models

Functional linear and single-index models are core regression methods in functional data analysis and are widely used methods for performing regression when the covariates are observed random functions coupled with scalar responses in a wide range of applications. In the existing literature, however, the construction of associated estimators and the study of their theoretical properties is invariably carried out on a case-by-case basis for specific models under consideration. In this work, we provide a unified methodological and theoretical framework for estimating the index in functional linear and single-index models; in the later case the proposed approach does not require the specification of the link function. In terms of methodology, we show that the reproducing kernel Hilbert space (RKHS) based functional linear least-squares estimator, when viewed through the lens of an infinite-dimensional Gaussian Stein's identity, also provides an estimator of the index of the single-index model. On the theoretical side, we characterize the convergence rates of the proposed estimators for both linear and single-index models. Our analysis has several key advantages: (i) we do not require restrictive commutativity assumptions for the covariance operator of the random covariates on one hand and the integral operator associated with the reproducing kernel on the other hand; and (ii) we also allow for the true index parameter to lie outside of the chosen RKHS, thereby allowing for index mis-specification as well as for quantifying the degree of such index mis-specification. Several existing results emerge as special cases of our analysis.